<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href="coqdoc.css" rel="stylesheet" type="text/css" /> <title>Flocq.Calc.Fcalc_digits</title> </head> <body> <div id="page"> <div id="header"> </div> <div id="main"> <h1 class="libtitle">Library Flocq.Calc.Fcalc_digits</h1> <div class="code"> </div> <div class="doc"> This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/ <div class="paragraph"> </div> Copyright (C) 2010-2013 Sylvie Boldo <br /> Copyright (C) 2010-2013 Guillaume Melquiond <div class="paragraph"> </div> This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. <div class="paragraph"> </div> This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. <div class="paragraph"> </div> <a name="lab13"></a><h1 class="section">Functions for computing the number of digits of integers and related theorems.</h1> </div> <div class="code"> <br/> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_Raux.html#"><span class="id" title="library">Fcore_Raux</span></a>.<br/> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#"><span class="id" title="library">Fcore_defs</span></a>.<br/> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_float_prop.html#"><span class="id" title="library">Fcore_float_prop</span></a>.<br/> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_digits.html#"><span class="id" title="library">Fcore_digits</span></a>.<br/> <br/> <span class="id" title="keyword">Section</span> <a name="Fcalc_digits"><span class="id" title="section">Fcalc_digits</span></a>.<br/> <br/> <span class="id" title="keyword">Variable</span> <span class="id" title="keyword">beta</span> : <a class="idref" href="Flocq.Core.Fcore_Zaux.html#radix"><span class="id" title="record">radix</span></a>.<br/> <span class="id" title="keyword">Notation</span> <a name="bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span> := (<a class="idref" href="Flocq.Core.Fcore_Raux.html#bpow"><span class="id" title="definition">bpow</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">e</span>).<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="Zdigits_ln_beta"><span class="id" title="lemma">Zdigits_ln_beta</span></a> :<br/> <span class="id" title="keyword">∀</span> <span class="id" title="var">n</span>,<br/> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> <a class="idref" href="Flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_Raux.html#ln_beta"><span class="id" title="definition">ln_beta</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="Flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#n"><span class="id" title="variable">n</span></a>).<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="ln_beta_F2R_Zdigits"><span class="id" title="lemma">ln_beta_F2R_Zdigits</span></a> :<br/> <span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span>, <a class="idref" href="Flocq.Calc.Fcalc_digits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> (<a class="idref" href="Flocq.Core.Fcore_Raux.html#ln_beta"><span class="id" title="definition">ln_beta</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="Flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#e"><span class="id" title="variable">e</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)%<span class="id" title="var">Z</span>.<br/> <br/> <span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Calc.Fcalc_digits.html#Fcalc_digits"><span class="id" title="section">Fcalc_digits</span></a>.<br/> </div> </div> <div id="footer"> <hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> </div> </div> </body> </html>
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